Provable Benefits of Complex Parameterizations for Structured State Space Models
This addresses a foundational open problem in machine learning for researchers and practitioners using SSMs like S4 and Mamba, though it is incremental as it takes a step towards resolution rather than providing a complete theory.
The paper tackles the theoretical explanation of why complex parameterizations in structured state space models (SSMs) are beneficial, proving that complex SSMs require lower dimensions and avoid exponentially large parameter values compared to real SSMs for expressing the same mappings.
Structured state space models (SSMs), the core engine behind prominent neural networks such as S4 and Mamba, are linear dynamical systems adhering to a specified structure, most notably diagonal. In contrast to typical neural network modules, whose parameterizations are real, SSMs often use complex parameterizations. Theoretically explaining the benefits of complex parameterizations for SSMs is an open problem. The current paper takes a step towards its resolution, by establishing formal gaps between real and complex diagonal SSMs. Firstly, we prove that while a moderate dimension suffices in order for a complex SSM to express all mappings of a real SSM, a much higher dimension is needed for a real SSM to express mappings of a complex SSM. Secondly, we prove that even if the dimension of a real SSM is high enough to express a given mapping, typically, doing so requires the parameters of the real SSM to hold exponentially large values, which cannot be learned in practice. In contrast, a complex SSM can express any given mapping with moderate parameter values. Experiments corroborate our theory, and suggest a potential extension of the theory that accounts for selectivity, a new architectural feature yielding state of the art performance.